Derek Goldrei

As its title promises, this textbook presents an introduction to the propositional calculus and to the predicate calculus. It is written in a clear but conversational manner that should make this subject readily accessible to any mathematically mature reader. The clear writing and good motivation should be appreciated by students using it as a text in a course as well as readers using it for independent study.

While the focus of Goldrei’s book is on the formalism of logic, he draws examples from mathematics to motivate and to enrich this material. This appears to be the greatest strength of his treatment of the subject. Examples come from algebra, number theory, geometry and topology, but the examples are explored without much risk of scaring away a student who lacks a good background in a few of these areas.

A second strength of this textbook is that it introduces the reader to a broad swath of concepts at the heart of the foundations of mathematics. After working through the book, the student should have a good understanding of semantics for both the propositional and predicate calculus, though paradoxically the author does not appear to introduce either of these specific terms. Still, the student of this text will come away with an understanding of models, some techniques for constructing them, and an appreciation for the existence of countable and non-standard models. At the same time the student will come away with some understanding of decidable theories and an appreciation for purely syntactical arguments.

However, the student will not be exposed to recursive functions or to Gödel’s incompleteness theorem. Perhaps Goldrei should consider writing a second volume to accompany this fine introduction.

Paul Cohen received his Ph.D. from the University of Illinois, was appointed as a Member of the Institute for Advanced Study by Kurt Gödel, and has taught at the University of Tennessee and at Lehigh University. He currently lives in Maine and is teaching at Colby College.